Filtrations associated with singularities

We fix a complex analytic normal singularity germ $(X,o)$ of dimension ≥ 2 and a (not necessarily irreducible) reduced Weil divisor (S,o)⊂ (X,o). The embedded resolution of the pair determines a multi-index filtration of the local ring OX,o, which measures the embedded geometry of the pair. Furthermore, from the (induced) resolution of $(S,o)$ we also consider a multi-index filtration associated with $(S,o)$. This latter one can be lifted to a filtration of OX,o too. The main result proves that the second filtration of OX,o can be realized as a `limit' filtration of the first one (if we blow up certain centers sufficiently many times).